Fractal and Fractional

13 pages, 2037 KiB

Open AccessArticle

Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation

by Melis Güneri and Mustafa Saltan

Fractal Fract. 2019, 3(1), 13; https://doi.org/10.3390/fractalfract3010013 - 25 Mar 2019

Cited by 3 | Viewed by 2978

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we [...] Read more.

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples. Full article

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13 pages, 284 KiB

Open AccessArticle

Some New Fractional Trapezium-Type Inequalities for Preinvex Functions

by Artion Kashuri, Erhan Set and Rozana Liko

Fractal Fract. 2019, 3(1), 12; https://doi.org/10.3390/fractalfract3010012 - 24 Mar 2019

Cited by 7 | Viewed by 2139

Abstract

In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out [...] Read more.

In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from the main results. Some applications regarding special means for different real numbers are provided as well. The ideas and techniques described in this paper may stimulate further research. Full article

9 pages, 364 KiB

Open AccessArticle

On the Fractal Langevin Equation

by Alireza Khalili Golmankhaneh

Fractal Fract. 2019, 3(1), 11; https://doi.org/10.3390/fractalfract3010011 - 13 Mar 2019

Cited by 13 | Viewed by 2731

Abstract

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle-

τ

Cantor set. The fractal mean square displacement of different random walks on the middle-

τ

Cantor set are presented. Fractal under-damped and over-damped [...] Read more.

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle-

τ

Cantor set. The fractal mean square displacement of different random walks on the middle-

τ

Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details. Full article

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8 pages, 239 KiB

Open AccessArticle

On Analytic Functions Involving the q-Ruscheweyeh Derivative

by Khalida Inayat Noor

Fractal Fract. 2019, 3(1), 10; https://doi.org/10.3390/fractalfract3010010 - 10 Mar 2019

Cited by 5 | Viewed by 2110

Abstract

In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied. Full article

16 pages, 4452 KiB

Open AccessArticle

Residual Power Series Method for Fractional Swift–Hohenberg Equation

by D. G. Prakasha, P. Veeresha and Haci Mehmet Baskonus

Fractal Fract. 2019, 3(1), 9; https://doi.org/10.3390/fractalfract3010009 - 07 Mar 2019

Cited by 51 | Viewed by 3946

Abstract

In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift–Hohenberg [...] Read more.

In this paper, the approximated analytical solution for the fractional Swift–Hohenberg (S–H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift–Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology. Full article

(This article belongs to the Special Issue 2019 Selected Papers from Fractal Fract’s Editorial Board Members)

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8 pages, 743 KiB

Open AccessArticle

The Fractal Calculus for Fractal Materials

by Fakhri Khajvand Jafari, Mohammad Sadegh Asgari and Amir Pishkoo

Fractal Fract. 2019, 3(1), 8; https://doi.org/10.3390/fractalfract3010008 - 06 Mar 2019

Cited by 13 | Viewed by 3947

Abstract

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal [...] Read more.

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using

F^{α}

-fractal calculus, in this paper, we derive exact

F^{α}

-differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility. Full article

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21 pages, 1079 KiB

Open AccessArticle

Fractal Image Interpolation: A Tutorial and New Result

by Chi Wah Kok and Wing Shan Tam

Fractal Fract. 2019, 3(1), 7; https://doi.org/10.3390/fractalfract3010007 - 23 Feb 2019

Cited by 5 | Viewed by 4146

Abstract

This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this [...] Read more.

This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this paper are based on the plain Iterative Function System (IFS) in spatial domain without additional transformation, where we believe that the benefits of additional transformation can be added onto the presented study without complication. Simulation results are presented to demonstrate the discussed techniques, together with the pros and cons of each techniques. Finally, a novel spatial domain interleave layer has been proposed to add to the IFS image system for improving the performance of the system from image zooming to interpolation with the preservation of the pixel intensity from the original low resolution image. Full article

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9 pages, 844 KiB

Open AccessBrief Report

Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

by Vance Blankers, Tristan Rendfrey, Aaron Shukert and Patrick D. Shipman

Fractal Fract. 2019, 3(1), 6; https://doi.org/10.3390/fractalfract3010006 - 20 Feb 2019

Cited by 9 | Viewed by 5205

Abstract

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form [...] Read more.

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form

x + τ y

for

x, y \in R

, and

τ^{2} = 1

but

τ \neq \pm 1

, are the natural number system in which to encode geometric properties of the Minkowski space

R^{1, 1}

. We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets. Full article

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9 pages, 270 KiB

Open AccessArticle

On q-Uniformly Mocanu Functions

by Rizwan S. Badar and Khalida Inayat Noor

Fractal Fract. 2019, 3(1), 5; https://doi.org/10.3390/fractalfract3010005 - 11 Feb 2019

Cited by 1 | Viewed by 2171

Abstract

Let f be analytic in open unit disc

E = {z : | z | < 1}

with

f (0) = 0

and

f^{^{'}} (0) = 1

. The q-derivative of f is defined by: [...] Read more.

Let f be analytic in open unit disc

E = {z : | z | < 1}

with

f (0) = 0

and

f^{^{'}} (0) = 1

. The q-derivative of f is defined by:

D_{q} f (z) = \frac{f (z) - f (q z)}{(1 - q) z}, q \in (0, 1), z \in B - {0},

where

B

is a q-geometric subset of

C

. Using operator

D_{q}

, q-analogue class

k - U M_{q} (α, β)

, k-uniformly Mocanu functions are defined as: For

k = 0

and

q \to 1^{-}

,

k -

reduces to

M (α)

of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases. Full article

15 pages, 489 KiB

Open AccessEditor’s ChoiceArticle

Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions

by Dimiter Prodanov

Fractal Fract. 2019, 3(1), 4; https://doi.org/10.3390/fractalfract3010004 - 25 Jan 2019

Cited by 5 | Viewed by 2805

Abstract

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under [...] Read more.

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erdélyi-Kober fractional differential and integral operators is demonstrated and discussed. Full article

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)

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6 pages, 911 KiB

Open AccessArticle

Study of Fractal Dimensions of Microcrystalline Cellulose Obtained by the Spray-Drying Method

by Michael Ioelovich

Fractal Fract. 2019, 3(1), 3; https://doi.org/10.3390/fractalfract3010003 - 24 Jan 2019

Cited by 7 | Viewed by 2351

Abstract

In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found [...] Read more.

In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found that increasing the average radius (R) of the formed MCC beads resulted in increased specific pore volume (P) and reduced apparent density (ρ). The dependences of P and ρ on the scale factor (R/r) can be expressed by power-law equations: P = P_o (R/r)^E−D_p and ρ = d (R/r)^D_d^−E, where the fractal dimensions D_p = 2.887 and D_d = 2.986 are close to the Euclidean dimension E = 3 for three-dimensional space; r = 3 nm is the radius of the cellulose nanocrystalline particles, P_o = 0.03 cm³/g is the specific pore volume, and d = 1.585 g/cm³ is the true density (specific gravity) of the CNP, respectively. With the increase in the size of the formed MCC beads, the order in the packing of the beads was distorted, conforming to theory of the diffusion-limited aggregation process. Full article

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2 pages, 127 KiB

Open AccessEditorial

Acknowledgement to Reviewers of Fractal Fract in 2018

by Fractal Fract Editorial Office

Fractal Fract. 2019, 3(1), 2; https://doi.org/10.3390/fractalfract3010002 - 16 Jan 2019

Viewed by 1769

Abstract

Rigorous peer-review is the corner-stone of high-quality academic publishing [...] Full article

11 pages, 340 KiB

Open AccessEditor’s ChoiceArticle

Regularized Integral Representations of the Reciprocal Gamma Function

by Dimiter Prodanov

Fractal Fract. 2019, 3(1), 1; https://doi.org/10.3390/fractalfract3010001 - 12 Jan 2019

Cited by 11 | Viewed by 3452

Abstract

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For [...] Read more.

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine’s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided. Full article

(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering 2018)

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Fractal Fract., Volume 3, Issue 1 (March 2019) – 13 articles

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